quasi
Posts:
12,048
Registered:
7/15/05


Re: Cardinality of turning wheel
Posted:
Mar 8, 2013 7:23 PM


netzweltler wrote: >quasi wrote: >> netzweltler wrote: >> >quasi wrote: >> >> >> >> Any infinite set of pairwise disjoint intervals on the real >> >> line is countably infinite since each interval contains a >> >> distinct rational number. >> > >> >t = 0 s: I am marking >> >[0, 0.5] #0, [0.5, 0.75] #1, [0.75, 0.875] #2, ... >> > >> >t = 0.5 s: I am marking >> >[0.5, 0.75] #0, [0.75, 0.875] #1, [0.875,0.9375] #2, ... >> > >> >t = 0.75 s: I am marking [0.75, 0.875] #0, [0.875, 0.9375] #1, >> >[0.9375, 0.96875] #2, ... >> >... >> >> >Which segments have been marked #0 after 1 s? >> >> I don't understand the mechanics or the intent of the above >> marking scheme. >> >> But I think you've fallen victim to a common fallacy. >> >> When trying to determine whether or not an infinite set is >> countable, a failed counting _doesn't_ disprove countability. > >I am not trying to prove or disprove countability of these >segments in [0, 1]. Remember, we have been discussing if the >choice of #1 (or #0 as you suggested) is arbitrary. Let me give >the answer to my question instead: All segments in [0, 1] have >been marked #0 at t = 1 s. So,there are no segments left to be >marked #1, #2, ..., are there?
I don't understand the intent of your marking scheme.
I thought the original question was, assuming finite, positive speeds, what is the cardinality of the set of revolutions of a wheel?
Answer: aleph0
In other words, the set of revolutions is countably infinite.
The idea is that each revolution corresponds to an interval of time on the real line (with arbitrary choice of the location of time t = 0), and that infinite set of nonoverlapping intervals on the real line is countably infinite.
Your attempt to match up intervals with markers in such a was as to leave some intervals unmarked has no impact on the original question.
For the purposes of determining countability, all we need is _some_ way of marking the intervals, not necessarily _your_ way, and not necessarily a way which matches the order of the markers with the order of the lefttoright placement of the intervals.
quasi

